Combinatorics of Cluster Varieties

聚类代数是用来描述高度对称数学对象的基本数学结构。在过去的十年中，星团结构在空间上的构建贯穿于表征理论和数学物理，最近在高能粒子物理的散射振幅理论中。该项目包括对几个关键簇代数的组合结构以及与之相关的几何空间的基础数学研究。本研究项目探讨了关于各种簇代数的组合结构的几个问题。这些聚类代数大多来自代数几何中的自然空间，如开放的正电荷变体，它们出现在分层的Grassmannian和广义Teichmuller空间中。项目的第一部分研究了在Grassmannian（和其他相关空间）中从一个簇到另一个簇变化时发生的Laurent多项式的系数。该项目的第二部分旨在确定环面上通过二聚体配置的正体细胞的参数化与这些细胞上作为图表的Plucker坐标值之间的关系。项目的第三部分调查簇品种的上同调，试图找到上同调环的表示是明确的和组合的，因为已经知道的坐标环的表示。

Cluster algebras are a fundamental mathematical structure employed to describe highly symmetric mathematical objects. In the last decade, cluster structures have been constructed on spaces throughout representation theory and mathematical physics, most recently in the theory of scattering amplitudes in high-energy particle physics. This project comprises fundamental mathematical research into the combinatorial structure of several crucial cluster algebras, and the geometric spaces related to them. This research project addresses several questions concerning the combinatorial structure of various cluster algebras. Most of these cluster algebras come from natural spaces in algebraic geometry, such as open positroid varieties, which occur in stratifying the Grassmannian and generalized Teichmuller spaces. The first part of the project investigates the coefficients of the Laurent polynomials that occur in changing from one cluster to another in the Grassmannian (and other related spaces). The second part of the project aims to determine the relation between the parametrizations of positroid cells by dimer configurations on tori and the values of Plucker coordinates as charts on those cells. The third part of the project investigates the cohomology of cluster varieties, trying to find presentations for the cohomology ring that are as explicit and combinatorial as the presentations already known for the coordinate rings.